Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347544 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62825 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,11)(4,8)(5,17)(6,9)(7,14)(10,19)(12,25)(13,24)(15,22)(16,29)(18,21)(20,27)(23,28)(26,32)(30,31)(33,34,38,42)(35,39,45,52)(36,40,47,54)(37,48,49,41)(43,55,57,50)(44,58,59,51)(46,60,61,53)(56,62,64,63);(1,3)(2,7)(4,12)(5,18)(6,13)(8,20)(9,21)(10,24)(11,17)(14,19)(15,26)(16,30)(22,31)(23,32)(25,29)(27,28)(33,35)(34,39)(36,43)(37,44)(38,45)(40,50)(41,51)(42,52)(46,56)(47,57)(48,58)(49,59)(53,62)(54,55)(60,63)(61,64);(1,4,6,15)(2,8,9,22)(3,12,13,26)(5,16,19,28)(7,20,21,31)(10,23,17,29)(11,25,24,32)(14,27,18,30)(33,36)(34,40)(35,43)(37,46)(38,47)(39,50)(41,53)(42,54)(44,56)(45,57)(48,60)(49,61)(51,62)(52,55)(58,63)(59,64);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30)(33,37,38,49)(34,41,42,48)(35,44,45,59)(36,46,47,61)(39,51,52,58)(40,53,54,60)(43,56,57,64)(50,62,55,63);(2,9)(3,13)(8,22)(10,17)(12,26)(14,18)(23,29)(27,30)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254761);